Relativizations of the of the Relativizations P =? DNP Question P =? DNP Question for the BSS Model for the BSS Model Christine Gaßner Christine Gaßner Greifswald Greifswald
The machines machines The Ljubljana 2009 gassnerc@uni-greifswald.de
The complexity complexity classes classes The Ljubljana 2009 gassnerc@uni-greifswald.de
The complexity complexity classes classes The Ljubljana 2009 gassnerc@uni-greifswald.de
The complexity complexity classes classes The Ljubljana 2009 gassnerc@uni-greifswald.de
The oracle oracle machines machines The Ljubljana 2009 gassnerc@uni-greifswald.de
A summary A summary Ljubljana 2009 gassnerc@uni-greifswald.de
A summary A summary Similarly to P Q ≠ P Q Q ≠ NP NP Q cp. CCA 2008 Q Q Similarly to P Q ≠ P Q Q ≠ NP NP Q cp. CCA 2008 Q Q Ljubljana 2009 gassnerc@uni-greifswald.de
A summary A summary cp. CCC 2009 cp. CCC 2009 Ljubljana 2009 gassnerc@uni-greifswald.de
A summary A summary Ljubljana 2009 gassnerc@uni-greifswald.de
Q ≠ Q Q Q An oracle Q Q with ≠ DN with P P R DNP P R An oracle R R Diagonalization techniques from Diagonalization techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only (only 0 and 1 as constants 0 and 1 as constants) ) Ljubljana 2009 gassnerc@uni-greifswald.de
Q ≠ Q Q Q An oracle Q Q with ≠ DN with P P R DNP P R An oracle R R Diagonalization techniques from Diagonalization techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only (only 0 and 1 as constants 0 and 1 as constants) ) BSS - BSS - only with 0 and 1 only with 0 and 1 Ljubljana 2009 gassnerc@uni-greifswald.de
Q ≠ Q Q Q An oracle Q Q with ≠ DN with P P R DNP P R An oracle R R Diagonalization techniques from Diagonalization techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only (only 0 and 1 as constants 0 and 1 as constants) ) BSS - BSS - only with 0 and 1 only with 0 and 1 Ljubljana 2009 gassnerc@uni-greifswald.de
Q ≠ Q Q Q An oracle Q Q with ≠ DN with P P R DNP P R An oracle R R Diagonalization techniques from Diagonalization techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only (only 0 and 1 as constants 0 and 1 as constants) ) BSS - BSS - only with 0 and 1 only with 0 and 1 Ljubljana 2009 gassnerc@uni-greifswald.de
Q ≠ Q Q Q An oracle Q Q with ≠ DN with P P R DNP P R An oracle R R Diagonalization techniques from Diagonalization techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only (only 0 and 1 as constants 0 and 1 as constants) ) BSS - BSS - only with 0 and 1 only with 0 and 1 Ljubljana 2009 gassnerc@uni-greifswald.de
Q ≠ Q Q Q An oracle Q Q with ≠ DN with P P R DNP P R An oracle R R Diagonalization techniques from Diagonalization techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only (only 0 and 1 as constants 0 and 1 as constants) ) BSS - BSS - only with 0 and 1 only with 0 and 1 V Important: V ≠ ∅ iff ≠ ∅ iff N i i N N W W i i rejects (0,..., 0) rejects i i (0,..., 0) ∈ ∈ ℝ ℝ n n i i Ljubljana 2009 gassnerc@uni-greifswald.de
Q ≠ Q Q Q An oracle Q Q with ≠ DN with P P R DNP P R An oracle R R Diagonalization techniques from Diagonalization techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only (only 0 and 1 as constants 0 and 1 as constants) ) BSS - BSS - only with 0 and 1 only with 0 and 1 V i V i ≠ ∅ ≠ ∅ can be satisfied can be satisfied Ljubljana 2009 gassnerc@uni-greifswald.de
Q ≠ Q Q Q An oracle Q Q with ≠ DN with P P R DNP P R An oracle R R Diagonalization techniques from Diagonalization techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only (only 0 and 1 as constants 0 and 1 as constants) ) BSS - BSS - only with 0 and 1 only with 0 and 1 V V ≠ ∅ can be satisfied ≠ ∅ can be satisfied i i since since i traversed by (0,..., 0) ∈ ∈ ℝ ℝ n the path of N N i Wi W ni traversed by (0,..., 0) i the path of i • is uniquely determined is uniquely determined • • of polynomial length of polynomial length • Ljubljana 2009 gassnerc@uni-greifswald.de
Wi ni W i rejects n (0,..., 0) ∈ ∈ ℝ ℝ N i N rejects (0,..., 0) i i Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only Diagonalization (only 0 and 1 as constants 0 and 1 as constants) ) W Wi N i N i : : i Ljubljana 2009 gassnerc@uni-greifswald.de
Wi ni W i rejects n (0,..., 0) ∈ ∈ ℝ ℝ N i N rejects (0,..., 0) i i Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only Diagonalization (only 0 and 1 as constants 0 and 1 as constants) ) Wi W N i N i : : i Only 0 and 1 as constants Only 0 and 1 as constants encoded by themselves encoded by themselves Ljubljana 2009 gassnerc@uni-greifswald.de
Wi ni W i rejects n (0,..., 0) ∈ ∈ ℝ ℝ N i N rejects (0,..., 0) i i Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only Diagonalization (only 0 and 1 as constants 0 and 1 as constants) ) Wi W N i N i : : i Only 0 and 1 as constants Only 0 and 1 as constants encoded by themselves encoded by themselves ⇒ ⇒ The polynomials are The polynomials are uniquely determined. uniquely determined. Ljubljana 2009 gassnerc@uni-greifswald.de
Wi ni W i rejects n (0,..., 0) ∈ ∈ ℝ ℝ N i N rejects (0,..., 0) i i Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only Diagonalization (only 0 and 1 as constants 0 and 1 as constants) ) Wi W N i N i : : i Only 0 and 1 as constants Only 0 and 1 as constants encoded by themselves encoded by themselves ⇒ ⇒ The polynomials are The polynomials are uniquely determined. uniquely determined. ⇒ ⇒ The path is uniquely The path is uniquely determined. determined. Ljubljana 2009 gassnerc@uni-greifswald.de
Q ≠ Q Q Q An oracle Q Q with ≠ DN with P P R DNP P R An oracle R R Diagonalization techniques from Diagonalization techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only (only 0 and 1 as constants 0 and 1 as constants) ) BSS - BSS - only with 0 and 1 only with 0 and 1 Ljubljana 2009 gassnerc@uni-greifswald.de
Wi ni W i rejects n ≠∅ iff (0,..., 0) ∈ ∈ ℝ ℝ i ≠∅ iff N N i V i rejects (0,..., 0) i V i Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only 0 and 1 as constants) Diagonalization (only 0 and 1 as constants) W Wi N i N i : : i 2 n ni length ≤ ≤ 2 length i ni ⇒∃ x ∈ {0, 1} Wi W i ) N i is not queried by N ( x x is not queried by ) ( i ⇒ V ≠∅ ⇒ i ≠∅ V i Ljubljana 2009 gassnerc@uni-greifswald.de
Q ≠ Q Q Q An oracle Q Q with ≠ DN with P P R DNP P R An oracle R R Diagonalization techniques from Diagonalization techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only (only 0 and 1 as constants 0 and 1 as constants) ) BSS - BSS - only with 0 and 1 only with 0 and 1 Important: N N ≙ N Q Q ≙ N W W ≙ N i i ≙ i i + + 1 1 N W i i W i i on (0,..., 0) on (0,..., 0) ∈ i i ∈ ℝ ℝ n n i i Ljubljana 2009 gassnerc@uni-greifswald.de
Q ≠ Q Q Q An oracle Q Q with ≠ DN with P P R DNP P R An oracle R R Diagonalization techniques from Diagonalization techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only (only 0 and 1 as constants 0 and 1 as constants) ) BSS - BSS - only with 0 and 1 only with 0 and 1 ⇒ ⇒ N i N W Wi +1 ≙ i +1 ≙ i N i N Q Q o o n n i ( ( 0 0 , , . . . . . . , , 0 0 ∈ ) ∈ ) ℝ n ℝ ni i Ljubljana 2009 gassnerc@uni-greifswald.de
Q on +1 ≙ Q Wi ni W n ≙ N (0,..., 0) ∈ ∈ ℝ ℝ N i N i N i +1 on (0,..., 0) i i i Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only 0 and 1 as constants) Diagonalization (only 0 and 1 as constants) Q : Q N i N : s µ , s s λ ≤ n n i s µ , λ ≤ i i +1 +1 ⇓ ⇓ ≙ ≙ ∈ W ... ∈ W i ? ... +1 ? i +1 ≙ ≙ ∈ W ... ∈ W i ? ... +1 ? i +1 Ljubljana 2009 gassnerc@uni-greifswald.de
Q ≠ Q Q Q An oracle Q Q with ≠ DN with P P R DNP P R An oracle R R Diagonalization techniques from Diagonalization techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only (only 0 and 1 as constants 0 and 1 as constants) ) BSS - BSS - only with 0 and 1 only with 0 and 1 ⇒ N ⇒ N i W Wi +1 ≙ i +1 ≙ N i N i Wi W i on on (0,..., 0) i (0,..., 0) ∈ ∈ ℝ ℝ n ni i Ljubljana 2009 gassnerc@uni-greifswald.de
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